The construction C \mapsto \widehat{C} = \mathbf{Fct}(C^{\text{op}}, \mathbf{Set}) is called the
colimit completion of
C because of the following
universal property: {{math_theorem|name=Proposition|math_statement=Let
C,
D be categories and assume
D admits small colimits. Then each functor \eta: C \to D factorizes as :C \overset{y}\longrightarrow \widehat{C} \overset{\widetilde{\eta}}\longrightarrow D where
y is the Yoneda embedding and \widetilde{\eta}: \widehat{C} \to D is a, unique up to isomorphism, colimit-preserving functor called the
Yoneda extension of \eta.}}
Proof: Given a presheaf
F, by the
density theorem, we can write F =\varinjlim y U_i where U_i are objects in
C. Then let \widetilde{\eta} F = \varinjlim \eta U_i, which exists by assumption. Since \varinjlim - is functorial, this determines the functor \widetilde{\eta}: \widehat{C} \to D. Succinctly, \widetilde{\eta} is the left
Kan extension of \eta along
y; hence, the name "Yoneda extension". To see \widetilde{\eta} commutes with small colimits, we show \widetilde{\eta} is a left-adjoint (to some functor). Define \mathcal{H}om(\eta, -): D \to \widehat{C} to be the functor given by: for each object
M in
D and each object
U in
C, :\mathcal{H}om(\eta, M)(U) = \operatorname{Hom}_D(\eta U, M). Then, for each object
M in
D, since \mathcal{H}om(\eta, M)(U_i) = \operatorname{Hom}(y U_i, \mathcal{H}om(\eta, M)) by the Yoneda lemma, we have: :\begin{align} \operatorname{Hom}_D(\widetilde{\eta} F, M) &= \operatorname{Hom}_D(\varinjlim \eta U_i, M) = \varprojlim \operatorname{Hom}_D(\eta U_i, M) = \varprojlim \mathcal{H}om(\eta, M)(U_i) \\ &= \operatorname{Hom}_{\widehat{C}}(F, \mathcal{H}om(\eta, M)), \end{align} which is to say \widetilde{\eta} is a left-adjoint to \mathcal{H}om(\eta, -). \square The proposition yields several corollaries. For example, the proposition implies that the construction C \mapsto \widehat{C} is functorial: i.e., each functor C \to D determines the functor \widehat{C} \to \widehat{D}. == Variants ==