I want to ask a stupid question. Let $I$ be an infinite set and suppose $i$ belongs to $I$. I wonder whether following morphisms exist in general:

Hom($A$,colim $B_i) \to$ lim Hom($A,B_i$) and

Colim Hom($A,B_i) \to$ Hom($A$,colim $B_i$)

What I know is: if we replace lim by infinite product and colim by infinite coproduct, it exists. But I am not sure in this general case above.