Does $\mbox{Hom}(\bullet,A)$ functor preserves pullbacks?

Question

I know that $\mbox{Hom}(A,\bullet)$ functor preserves pullback (Hom-functor preserves pullbacks) but what could we say about the contravariant functor $\mbox{Hom}(\bullet,A)$?

Answer 1

Depends on what you mean by that. It preserves limits (in particular pullbacks) as a functor $\mathcal{A}^{\operatorname{op}} \to \mathsf{Set}$. But this means as a contravariant functor $\mathcal{A} \to \mathsf{Set}$ it sends colimits to limits (in particular pushouts to pullbacks).

(here $\mathcal{A}$ is your ambient category)