Exercise 2.16

Proof. By the universal property of the direct sum (Lemma ?? ), we get a unique homomorphism $\bar{\alpha }$ fitting into the commutative diagram

$$$$ ⊕ i M i μ ι α ι α α i i i j j ¯j N" />

where ${\iota }_i$ are the canonical inclusions $M_i\to {\oplus <!--FUNCTION\; APPLICATION-->}_i{M}_i$. By the commutativity of the diagram, $\bar{\alpha }(x_i)=\bar{\alpha }({\mu }_{ij}(x_i))$, hence $\bar{\alpha }$ factors uniquely through ${\oplus <!--FUNCTION\; APPLICATION-->}_i{M}_i∕D=\underset{\to }{\lim <!--FUNCTION\; APPLICATION-->}{M}_i$ where $D$ is as before, i.e., there exists unique $\alpha :\underset{\to }{\lim <!--FUNCTION\; APPLICATION-->}{M}_i\to N$ such that $\bar{\alpha }=\alpha \circ \pi$, where $\pi :{\oplus <!--FUNCTION\; APPLICATION-->}_i{M}_i\to \underset{\to }{\lim <!--FUNCTION\; APPLICATION-->}{M}_i$ is the quotient map [?, Thm.  $14.1.6(b)$]. Defining ${\mu }_i=\pi \circ {\iota }_i$, we have that ${\alpha }_i=\alpha \circ {\mu }_i$ by the commutativity of the diagram

$$$$ ⊕ i M i N j μ α απα i i i ¯ lim → M i " />

for each $i$. The direct limit is characterized up to isomorphism by this property since if $M^{\prime }$ also satisfied this property, then the universal property gives homomorphisms $M^{\prime }\to \underset{\to }{\lim <!--FUNCTION\; APPLICATION-->}{M}_i$ and $\underset{\to }{\lim <!--FUNCTION\; APPLICATION-->}{M}_i\to M^{\prime }$ such that their composition must be the identity (where we use the universal property again), hence they are isomorphic. □