In the exercises of Atiyah-Macdonald, it defines a direct system of $R$-modules over a directed set $I$ as a collection of $R$-modules $M_i$ and homomorphisms $\phi_{ij} : M_i \to M_j$ for each $i\leq j$, such that $\phi_{ii}$ is the identity map of $M_i$, and $\phi_{jk} \circ \phi_{ij} = \phi_{ik}$ for $i\leq j \leq k$.
And then it constructs the direct limit $M$ of the system and proves some of its properties. My question is, I can't see why the condition $\phi_{ii}=1_{M_i}$ is needed in any of this. It does seem like a natural condition, but would anything go wrong without that condition?
This is necessary because a directed set is just a preorder, not a partial order; i.e., it’s not required to be antisymmetric, as a partial order is.
If a directed set were defined as a partial order in which any two elements have an upper bound, you could just leave out the homomorphisms $\phi_{ii}$. Later in the construction, where they form the quotient over a submodule generated by all elements of the form $x_i-\phi_{ij}(x_i)$ where $i\le j$ and $x_i\in M_i$, you’d then have to write $i\lt j$ instead. If you allow $i=j$ with $\phi_{ii}$ not the identity on $M_i$, you quotient out some of the structure of $M_i$, rather than just quotienting out the difference between corresponding elements in different $M_i$ as intended. The idea of a direct limit is that in a sense it contains each of the objects, and that wouldn’t be the case then.
Because a directed set is just a preorder, not a partial order, there may be $i\ne j$ with $i\le j$ and $j\le i$, so to make the whole thing work you need to include $\phi_{ji}\circ\phi_{ij}=\phi_{ii}$, and if you don’t require it to be the identity, you end up quotienting out part of the structure.