Consider the following diagram:
Given that $h \circ f = k$ and $l \circ h = g$, prove that $ g \circ f = l \circ k $
Answer:
page-35, An introduction to Category Theory, Harold Simmons
I am a bit confused, doesn't the first diagram itself have the proof? It is pretty clear that moving by $k$ then by $l$ gives same thing as moving by $f$ and $g$ to me. So, what more of an idea is put in the step wise proof?
I think you are being falsely reassured by the picture of the diagram, which makes things seem like they are obvious. But imagine you are given a group $G$, and elements $f,g,h,k,l\in G$ such that $h\cdot f=k$ and $l\cdot h =g$. Now you want to prove that $g\cdot f = l\cdot k$. Of course, it's easy: $$g\cdot f=(l\cdot h)\cdot f = l\cdot (h\cdot f) = l\cdot k,$$ but it's clear that it requires this computation to give the proof. You would not get away with saying "it's just clear".
The visual dimension of diagrams is extremely useful, but you should not forget that in the end it's just a shortcut for a lot of formal equations, and that any "visual reasoning" is justified by computations. The computation you're looking at is basically saying that in a diagram, as long as all triangles commute, then everything commutes.