The fibrational slice of a 2-category $K$ over an object $X$ is the homwise-full sub-2-category $\mathrm{Fib}(X)$ of the slice 2-category $K/X$ whose objects are fibrations over $X$ and whose morphisms are morphisms of fibrations. Likewise, we have the opfibrational slice $\mathrm{Opf}(X)$ consisting of opfibrations.
The fibrational and opfibrational slices in a 2-category often play the role of the ordinary slice categories of a 1-category, replacing the ordinary slice 2-category. On this whole page we assume that $K$ has finite limits.
The 2-category $\mathrm{Opf}(X)$ is monadic over $K/X$. The relevant monad on $K/X$ takes $p:A\to X$ to the comma object $(p/{1}_{X})$, or equivalently the pullback $A{\times}_{X}{X}^{2}$. It is lax-idempotent, so a morphism $p:A\to X$ is an opfibration if and only if $A\to (p/{1}_{X})$ has a left adjoint with invertible counit in $K/X$. Likewise, $p$ is a fibration iff $A\to ({1}_{X}/p)$ has a right adjoint with invertible unit in $K/X$.
Since the fibrational slices are monadic over $K/X$, they inherit all limits from it. It follows that a fibration is a discrete object in $\mathrm{Fib}(X)$ iff it is discrete in $K/X$. These are unsurprisingly called discrete fibrations; we write $\mathrm{DFib}(X)$ for the category of such. Every morphism in $K/X$ between discrete fibrations is a morphism of fibrations; thus $\mathrm{DFib}(X)$ is a full subcategory of both $\mathrm{Fib}(X)$ and $K/X$.
Any pullback of an (op)fibration is again an (op)fibration. Therefore, any morphism $f:Y\to X$ induces a pullback functor ${f}^{*}:\mathrm{Fib}(X)\to \mathrm{Fib}(Y)$, which restricts to a functor ${f}^{*}:\mathrm{DFib}(X)\to \mathrm{DFib}(Y)$, and dually. Regarding the existence of adjoints to these functors, see comprehensive factorization and exponentials in a 2-category.
Any morphism with groupoidal codomain is a fibration and opfibration. Therefore, if $X$ is groupoidal, $\mathrm{Fib}(X)\simeq K/X\simeq \mathrm{Opf}(X)$. In particular, for (2,1)-categories and thus also for 1-categories, the fibrational slices are no different from the ordinary slices.
Central for us is the following fact, which would be false if we replaced $\mathrm{Opf}(X)$ by $K/X$. It underlies the inheritance of all sorts of structure by fibrational slices, such as regularity, coherency, extensivity, and exactness.
A morphism in $\mathrm{Opf}(X)$ is ff in $\mathrm{Opf}(X$) iff its underlying morphism in $K$ is ff.
Suppose that $a:A\to X$ and $b:B\to X$ are opfibrations, and that $fmapsA\to B$ is a morphism in $\mathrm{Opf}(X)$ which is ff in $K$. Then since $\mathrm{Opf}(X)\to K$ is homwise faithful, $f$ is clearly faithful in $\mathrm{Opf}(X)$. And given a 2-cell $\alpha :h\to k:T\rightrightarrows B$ in $\mathrm{Opf}(X)$, since $f$ is ff in $K$, we must have $\alpha =f\beta $ for some 2-cell $\beta :j\to l:T\rightrightarrows A$. But then $a\beta \cong bf\beta =b\alpha $, so $\beta $ must also be a 2-cell in $\mathrm{Opf}(X)$.
Conversely, suppose that $f$ is ff in $\mathrm{Opf}(X)$, and that we have 2-cells $\alpha ,\beta :x\rightrightarrows y:T\rightrightarrows A$ in $K$ such that $f\alpha =f\beta $. Then since $bf\cong a$ we have $aalph=a\beta $; call this 2-cell $\xi :ax\to ay$. Since $a$ and $b$ are opfibrations and $f$ is a map of opfibrations, we have an opcartesian 2-cell $x\to {\xi}_{!}(x)$ lying over $\xi $ such that $fx\to f{\xi}_{!}(x)$ is also opcartesian. Then $\alpha $ and $\beta $ both factor through $x\to {\xi}_{!}(x)$ to give 2-cells $\gamma ,\delta :{\xi}_{!}(x)\rightrightarrows y$ in $\mathrm{Opf}(X)$ whose images under $f$ are equal. Since $f$ is faithful in $\mathrm{Opf}(X)$, we have $\gamma =\delta $, and hence $\alpha =\beta $. Thus, $f$ is faithful in $K$. The proof of fullness is analogous.
Therefore, if $A\to X$ is an (op)fibration in $K$, we have ${\mathrm{Sub}}_{\mathrm{Opf}(K)}(A)\subset {\mathrm{Sub}}_{K}(A)$, although in general the inclusion is proper. Of course, the dual result about $\mathrm{Fib}(X)$ is also true.
It is well-known that a composite of fibrations is a fibration. Moreover:
A morphism in $\mathrm{Fib}(X)$ is a fibration in the 2-category $\mathrm{Fib}(X)$ iff its underlying morphism in $K$ is a fibration.
This is a standard result, at least in the case $K=\mathrm{Cat}$, and is apparently due to Benabou. References include:
Therefore, for any fibration $A\to X$ in $K$ we have ${\mathrm{Fib}}_{K}(A)\simeq {\mathrm{Fib}}_{{\mathrm{Fib}}_{K}(X)}(A\to X)$, and similarly for opfibrations. This is a fibrational analogue of the standard equivalence $K/A\simeq (K/X)/(A\to X)$ for ordinary slice categories. It also implies that any morphism between discrete fibrations over $X$ is itself a (discrete) fibration in $K$, since in $\mathrm{Fib}(X)$ it has a discrete (hence groupoidal) target and thus is a fibration there.
We will also need the corresponding result for mixed-variance iterated fibrations, which seems to be less well-known. First we recall:
A span $A\stackrel{p}{\leftarrow}E\stackrel{q}{\to}B$ is called a (two-sided) fibration from $B$ to $A$ if
$q$ is an opfibration and $p$ takes $q$-opcartesian 2-cells to isomorphisms,
$p$ is a fibration and $q$ takes $p$-cartesian 2-cells to isomorphisms, and
for any $e:X\to E$, and any square
of 2-cells, $({\alpha}^{*}e){\beta}_{!}\to {\alpha}^{*}(e{\beta}_{!})$ is an isomorphism.
Such two-sided fibrations in $\mathrm{Cat}$ correspond to functors $B\times {A}^{\mathrm{op}}\to \mathrm{Cat}$. The third condition corresponds precisely to the “interchange” equality $(\beta ,1)(1,\alpha )=(1,\alpha )(\beta ,1)$ in $B\times {A}^{\mathrm{op}}$. We write $\mathrm{Fib}(B,A)$ for the 2-category of two-sided fibrations from $B$ to $A$.
A span $A\stackrel{p}{\leftarrow}E\stackrel{q}{\to}B$ is a two-sided fibration from $B$ to $A$ if and only if
Recall that the projection $A\times B\to A$ is a fibration (and also an opfibration, although that is irrelevant here), and the cartesian 2-cells are precisely those whose component in $B$ is an isomorphism. Therefore, saying that $(p,q)$ is a morphism in $\mathrm{Fib}(A)$, i.e. that it preserves cartesian 2-cells, says precisely that $q$ takes $p$-cartesian 2-cells to isomorphisms.
Now, by the remarks above, $q$ is an opfibration in $K$ iff $E\to (q/{1}_{B})$ has a left adjoint with invertible counit in $K/B$, and $(p,q)$ is an opfibration in $\mathrm{Fib}(A)$ iff $E\to ((p,q)/{1}_{A\times B})$ has a left adjoint with invertible counit in $\mathrm{Fib}(A)/(A\times B)$. Of crucial importance is that here $((p,q)/{1}_{A\times B})$ denotes the comma object calculated in the 2-category $\mathrm{Fib}(A)$, or equivalently in $K/A$, and it is easy to check that this is in fact equivalent to the comma object $(q/{1}_{B})$ calculated in $K$.
Therefore, $(p,q)$ is an opfibration in $\mathrm{Fib}(A)$ iff $q$ is an opfibration in $K$ and the left adjoint of $E\to (q/{1}_{B})$ is a morphism in $\mathrm{Fib}(A)$. It is then easy to check that this left adjoint is a morphism in $K/A$ iff $p$ inverts $q$-opcartesian arrows, and that it is a morphism of fibrations iff the final condition in Definition 1 is satisfied.
In particular, we have $\mathrm{Fib}(B,A)\simeq {\mathrm{Opf}}_{\mathrm{Fib}(A)}(A\times B)$. By duality, $\mathrm{Fib}(B,A)\simeq {\mathrm{Fib}}_{\mathrm{Opf}(B)}(A\times B)$, and therefore ${\mathrm{Fib}}_{\mathrm{Opf}(B)}(A\times B)\simeq {\mathrm{Opf}}_{\mathrm{Fib}(A)}(A\times B)$, a commutation result that is not immediately obvious.
It follows that the 2-categories $\mathrm{Fib}(B,A)$ of two-sided fibrations also inherit any properties that can be shown to be inherited by the “one-sided” fibrational slices $\mathrm{Fib}(X)$ and $\mathrm{Opf}(X)$. Thus, we will usually concentrate on the latter, although two-sided fibrations will make an appearance in our treatment of duality involutions.