Consider the infinite diagrams: $$ C_1\to C_2 \to \cdots \to C_n \to \cdots $$ $$ D_1\to D_2 \to \cdots \to D_n \to \cdots $$ in some category, and suppose both colimits exist.
How do I check if the colimits are isomorphic? Is there a standard way, or even a definition?
To prove two (co-)limits are isomorphic, we usually show that one (co-)limit satisfies the universal property of the other one and vice versa.