Let $A,B,C,D$ be groups. Let $f:A\to B$ and $f':C\to D$ be a group homomorphism. Let $f\times f': A\times C\to B\times D$ be a homomorphism given by $(a,c)\to (f(a),f'(c))$.
Is it true that $ker(f\times f')=kerf\times kerf'$ and $\#coker(f\times f')=\#coker(f)\times \#coker(f')$ ?
I think this is clear because $ker(f\times f')=\{(a,b)\in A\times C\mid f(a)=0, f(b)=0\}=kerf\times kerf'$ and $im(f\times f')=\{(f(a),f(c))\mid (a, b)\in A\times C\}=imf\times imf'$.
I think this is almost definition, but I have never seen this statement in any text books, so I don't have confident.