Def: If X+Y is the coproduct of X and Y, let inj(X, X+Y) and inj(Y,X+Y) be the associated injection arrows.

Def: If X*Y is the product of X and Y, let pro(X*Y,X) and pro(X*Y, Y) be the associated projection arrows.

Def: If aXY is an arrow from X to Y and aYZ is an arrow from Y to Z, then let

comp(aYZ, aXY)

be the composition of aYZ and aXY. comp(aYZ, aXY) is an arrow from X to Z.

Notation: If you have two arrows f:Z->X, and g:Z->Y, the let be the unique arrow from Z to X*Y such that

comp( pro(X*Y,X) , ) = f and
comp( pro(X*Y,Y) , ) = g. (The existence of comes from the definition of X*Y.)

Notation: If you have two arrows f:X->Z, and g:Y->Z, the let [f,g] be the unique arrow from X+Y to Z such that

comp( [f,g], inj(X,X+Y) ) = f and
comp( [f,g], inj(Y,X+Y) ) = g. (The existence of [f,g] comes from the definition of X+Y.)

1) Let a_1 = comp( inj(B, B+C), pro(A*B, B). a_1: A*B -> B+C.
2) Let a_2 = comp( inj(B, B+C), pro(A*C, C). a_2: A*C -> B+C.
3) Let a_3 = [a_1, a_2]. a_3:(A*B)+(A*C) -> B+C.
4) Let a_4 = [pro(A*B,A), pro(A*C,A)]. a_4:(A*B)+(A*C) -> A.
5) Let a_5 = . a_5:(A*B)+(A*C) -> A*(B+C).

a_5 is jagr2808’s arrow.

(I am not a category theorist, so this notation may be totally non-standard, or maybe I saw it somewhere.)