Category Theory – Understanding Limits and Colimits

Question

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.

Answer 1

For any diagram $B_i$ and an object $A$ in a category, there are natural maps of sets:

1. colim Hom($A,B_i) \to$ Hom($A$, colim $B_i$)
2. colim Hom($B_i,A) \to$ Hom(lim $B_i, A$)

These maps need not be isomorphisms, in general (neither even when the diagram is filtered, nor when it is finite). Nor are they isomorphisms for infinite products and coproducts, in general (for finite products and coproducts in an additive category they are isomorphisms, though).

Besides, for any diagram $B_i$ and an object $A$ there are natural isomorphisms of sets:

1. Hom($A$, lim $B_i$) = lim Hom($A,B_i$)
2. Hom(colim $B_i, A$) = lim Hom($B_i,A$)

These isomorphisms hold for any diagram (it does not have to be filtered, nor does it have to be finite). Actually, they hold by the definition of lim and colim.