Abstract Algebra I · Quotient groups and direct products
Build a map into a direct product
Problem
Given homomorphisms $f:K\to G$ and $g:K\to H$, prove that $\langle f,g\rangle:K\to G\times H$, $x\mapsto(f(x),g(x))$, is the unique homomorphism whose coordinate projections are $f$ and $g$.
Hint
Multiplicativity follows coordinatewise from multiplicativity of $f$ and $g$.
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