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The Yoneda extension of a functor $F : C \to D$ is extension along the Yoneda embedding $Y : C \to [C^{op},Set]$ of its domain category to a functor
The Yoneda extension exhibits the presheaf category $PSh(C)$ as the free cocompletion of $C$.
For $C$ a small category and $F : C \to D$ a functor, its Yoneda extension
is the left Kan extension $Lan_Y F : [C^{op}, Set] \to D$ of $F$ along the Yoneda embedding $Y$:
Often it is of interest to Yoneda extend not $F : C \to D$ itself, but the composition $Y \circ F : C \to [D^{op}, Set]$ to get a functor entirely between presheaf categories
This is in fact a left adjoint to the direct image or restriction functor $F_* : [D^{op}, Set] \to [C^{op}, Set]$ which maps $H \mapsto H \circ F$; see restriction and extension of sheaves.
Recalling the general formula for the left Kan extension of a functor $F : C \to D$ through a functor $p : C \to C'$
one finds for the Yoneda extension the formula
(Recall the notation for the comma category $(Y,A) := (Y, const_A)$ whose objects are pairs $(U \in C, (Y(U) \to A) \in [C^{op}, Set] )$.
For the full extension $\hat F : [D^{op}, Set] \to [C^{op}. Set]$ this yields
Here the first step is from above, the second uses that colimits in presheaf categories are computed objectwise and the last one is again using the Yoneda lemma.
The restriction of the Yoneda extension to $C$ coincides with the original functor: $\tilde F \circ Y \simeq F$.
The Yoneda extension commutes with small colimits in $C$ in that for $\alpha : A \to C$ a diagram, we have $\tilde F (colim (Y \circ \alpha)) \simeq colim F \circ \alpha$ .
Moreover, $\tilde F$ is defined up to isomorphism by these two properties.