I want to ask a stupid question. I wonder whether following morphism exists in general 

Let I be an infinite set. i belongs to I 

Hom(A,colimBi)--->limHom(A,Bi) and 

Colim(A,B1)---->Hom(A,colimBi)

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