Exercise 2.18

Let $M=(M_i,{\mu }_{\mathit{ij}})$, $N=(N_i,{\nu }_{\mathit{ij}})$ be direct systems of $A$-modules over the same directed set. Let $M$, $N$ be the direct limits and ${\mu }_i:M_i\to M$, ${\nu }_i:N_i\to N$ the associated homomorphisms.

A homomorphism $\Phi :M\to N$ is by definition a family of $A$-module homomorphisms ${\phi }_i:M_i\to N_i$ such that ${\phi }_j\circ {\mu }_{\mathit{ij}}={\nu }_{\mathit{ij}}\circ {\phi }_i$ whenever $i\le j$. Show that $\Phi$ defines a unique homomorphism $\phi =\underset{\to }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{\phi }_i:M\to N$ such that $\phi \circ {\mu }_i={\nu }_i\circ {\phi }_i$ for all $i\in I$.