In 2-categorical algebra, there are many kinds of morphism between objects: e.g. strict morphisms, pseudo morphisms, lax morphisms, colax morphisms, etc. Typically it is the pseudo and (co)lax morphisms that arise in practice (for instance, we are rarely interested in strict monoidal functors), but the strict morphisms tend to be simpler to work with.

In many situations, we can study the weaker kinds of morphism using the strict kinds of morphism using a weak morphism classifier. A weak morphism classifier for an object $A$ is an object $A'$ such that weak morphisms $A \to B$ are in natural bijection with strict morphisms $A' \to B$.

Dually, a weak morphism coclassifier is an object $B'$ such that weak morphisms $A \to B$ are in natural bijection with strict morphisms $A \to B'$.

Definition

For algebras of 2-monads

Let $T$ be a 2-monad and denote by $T-Alg_s$ the 2-category of strict algebras? and strict morphisms for $T$. Denote by $T-Alg_w$ the 2-category of strict algebras? and $w$-weak morphisms for $T$ (where $w$ is pseudo, lax, colax, etc.). There is an identity-on-objects2-functor:

$T-Alg_s \to T-Alg_w$

If this admits a left 2-adjoint $(-)' : T-Alg_w \to T-Alg_s$, we call this the $w$-weak morphism classifier for $T$. It if admits a right 2-adjoint $T-Alg_w \to T-Alg_s$, we call this the $w$-weak morphism coclassifier for $T$.

Weak morphism classifiers exist if and only if certain codescent objects exist in $T-Alg_s$.