From Mac Lane's Category theory:
Can someone explain why the box in red in the picture below is true?
By the first line in the proof, the diagram shown to be commutative is really:
$$S(1_{b'}, g) \circ S(f, 1_c) = S(f, 1_{c'}) \circ S(1_b, g)$$
But the equation to be proven is $M_{b}'g \circ L_c f = L_{c'} f \circ M_b g$, which when applying the functors and bifunctors of the proposition means we have to prove:
$$S(b', g) \circ S(f, c) = S(f, c') \circ S(b, g)$$
Why exactly are these arrows equal?
If $f : b \rightarrow b'$ then $S(f,c) $ and $S(f, 1_c)$ have the same domain and codomain but why are they the same arrow?
The notation $S(f,c)$ is a shorthand for $S(f,1_c)$, since $c$ is used interchangeably with its identity morphism, as mentioned in the first line of the proof.