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Understanding the definition of equality

Suppose $f:A \rightarrow B$ and $g:B \rightarrow C$, and $g \circ f$ is undefined at the $a \in A$. Does it then mean that $$f \equiv g?$$
Furthermore, if it is not true what are the conditions under which it is true?

A:

The concept of “definition of a function” gets pretty handwavy and unclear with non-uniform sequences. In particular, you have a sort of weak version of (the uniformization of) $\mathbf{Set}$ that is a set $(A,f)$ equipped with a map $f:A\to \mathbf{Set}$ such that all points in $A$ are sent to points in $\mathbf{Set}$ and every element of $A$ has at most one preimage in $\mathbf{Set}$. So if we have a function $f:A\to B$ and $g:B\to C$ and $g\circ f$ is undefined at some element in $A$, it does not mean that $f\equiv g$. What it means is that we can’t map some element of $A$ to some element of $B$, so we haven’t defined $g\circ f$ as a map $A\to C$ in the sense that if we have two elements in $A$ that are sent to the same element in $B$, then we have an element in $C$ that is sent to this element.
The only example we can readily see of this is $f\equiv \operatorname{id}_B$ and $g\equiv\operatorname{id}_C$. While $f\equiv \operatorname{id}_B$ is not a function in the sense of the above definition, since any two elements are sent to the same element, it is a function if we let $g$ be any map. So in this specific case, $f\equiv g$.
On the other hand, you may think that \$f\equ

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