I am struggling to understand the definition of a final functor given on page 217 of MacLane's "Categories for the Working Mathematician." It says that a functor $L \colon J' \rightarrow J$ is final if for each $k \in J$ the comma category $(k/L)$ is nonempty and connected. Let's suppose that $L$ is an inclusion, and $J$ is the pushout data $$ B \leftarrow A \rightarrow C $$ If $L$ is simply the identity, it does not seem to satisfy the conditions of being final. That is, the comma category under the $A$ is not connected as far as I can tell. But surely the identity functor is final?
I'm sure I am misunderstanding something and would appreciate if someone could set me straight.
One way i build intuition is from when considering colimits indexed by directed posets.
If $P$ is some directed poset and $P'\subseteq P$ is some subset. Then the inclusion $i:P'\to P$ is a final functor if for all $p \in P$ there exists some $p'\in P'$ such that $p\leq p'$. Hence it is sufficient to compute colimits indexed by $P'$.
This plays nice with my intuition since in nice situations such colimits should be fought of as unions in which thise should be very clear.