I have a very basic question which is driving me nuts.
I was trying to show that an equalizer is always monic. I ended up doing something like in In Every equalizer is monic, namely:
Suppose $i : e \rightarrow a$ equalizes $f,g : a \rightarrow b$. Suppose $i \circ j = i \circ l$ where $j,l : c \rightarrow e$. Since $$f \circ (i \circ j) = (f \circ i) \circ j = (g \circ i) \circ j = g \circ (i \circ j)$$ there exists a unique $k : c \rightarrow e$ such that $i \circ k = i \circ j$. Hence, $k = j$. Since $i \circ l = i \circ j$, $k = l$. Hence, $j = l$ and $i$ is monic.
But there is something here that I do not get: I agree that there is a "unique $k : c \rightarrow e$ such that $i \circ k = i \circ j$. Hence, $k = j$." I also agree that there is a unique $k' : c \rightarrow e$ such that $i \circ k = i \circ l$. Hence, $k' = l$. But I do not agree that $k=k'$ necessarily. What am I missing? I do not think that got the definition of equalizer wrong...
There is a unique $k$ such that $i\circ k = i\circ j$. Thus $k=j$.
There is a unique $k'$ such that $i\circ k' = i\circ l$. But $i\circ l = i\circ j$, so $i\circ k' = i\circ j$ so $k'=k$ (unicity of $k$). So $i\circ k = i\circ l$ so $k=l$ (unicity of $k$ again). Therefore $j=k=l$, $j=l$.