I'm working through Arkowitz's Introduction to Homotopy Theory for self-study. In the beginning of chapter 5 section 2, we are aiming for a proof of (a simplified version of) the Universal Coefficient Theorem for (homotopical) cohomology (where $H^n(X;G)\equiv[X,K(G,n)]$ with $K(G,n)$ is an Eilenberg-MacLane space). Along the way, we're asked to prove the following.
Question
For a space $X$ and an abelian group $G$, we define a homomorphism $\alpha = \alpha_G : H^n(X) \otimes G \rightarrow H^n(X;G)$ as follows. Given $[f] \in H^n(X) = [X, K(\mathbb{Z},n)]$ and $\gamma \in G$, then $\gamma$ induces a homomorphism $\theta_\gamma: \mathbb{Z} \rightarrow G$ defined by $\theta_\gamma(1)=\gamma$. By earlier lemmas, we know that $\theta_{\gamma}$ determines a unique homotopy class $[\phi_\gamma] \in [K(\mathbb{Z},n),K(G,n)]$ given by $\phi_\gamma* = \theta_\gamma:\pi_n(K(\mathbb{Z},n))\rightarrow\pi_n(K(G,n))$. We set $\alpha([f]\otimes \gamma)=[\phi_\gamma f]$. Show $\alpha$ is a well-defined homomorphism by showing that
Part 1: $\phi_\gamma(f+f^\prime)\simeq\phi_\gamma(f) + \phi_\gamma(f^\prime)$
Part 2: $\phi_{\gamma + \gamma^\prime} (f)\simeq\phi_{\gamma}(f) + \phi_{\gamma^\prime}(f) $
Thoughts
One thing about this book is that it forces the reader, at key moments, to go back and review many of the earlier concepts/proofs to make sure that they are solid before proceeding. I believe I have a straightforward solution to Part 2, but remain stuck on Part 1.
For Part 1, maybe I should begin by replacing $K(\mathbb{Z},n)$ with $\Omega K(\mathbb{Z},n+1)$ and replace $K(G,n)$ with $\Omega K(G,n+1)$, just so that the addition operation is apparent. From here some ideas occur, but none have panned out so far.
To close this out, after reading the comment and some further discussion/review, I'll sketch a proof.
We require a Lemma (5.2.1) from the text
Lemma
If $G$ and $H$ are abelian groups and $f,g: K(G,n)\rightarrow K(H,n)$ are two maps, then $f_*=g_*:\pi_n(K(G,n))=G\rightarrow\pi_n(K(H,n))=H, n\geq1$, if and only if $f\simeq g$.
Proof of Part 1
We begin by observing that $$\phi_\gamma \circ(f+f^\prime)\equiv \phi_\gamma \circ\mu_{K(\mathbb{Z},n)}\circ (f\times f^\prime) \circ \Delta$$ where $\mu_{K(\mathbb{Z},n)}$ is the multiplication in $K(\mathbb{Z},n)$ and $\Delta$ is the diagonal map. Also
$$\phi_\gamma f+\phi_\gamma f^\prime\equiv \mu_{K(G,n)} \circ (\phi_\gamma f\times \phi_\gamma f^\prime) \circ \Delta= \mu_{K(G,n)} \circ (\phi_\gamma \times \phi_\gamma) \circ (f\times f^\prime) \circ \Delta.$$
So it will suffice to show that
$$\phi_\gamma \circ\mu_{K(\mathbb{Z},n)} \simeq \mu_{K(G,n)} \circ (\phi_\gamma \times \phi_\gamma): K(\mathbb{Z\times Z},n)\rightarrow K(G,n). $$
We note that $K(\mathbb{Z\times Z},n) \simeq K(\mathbb{Z},n) \times K(\mathbb{Z},n)$. For $(z, z^\prime) \in \pi_n(K(\mathbb{Z},n)) \times \pi_n(K(\mathbb{Z},n))$:
$$(\phi_\gamma \circ\mu_{K(Z,n)})_* [(z,z^\prime)]= \theta_\gamma [z + z^\prime] = \theta_\gamma [z] + \theta_\gamma [z^\prime] = (\mu_{K(G,n)} \circ (\phi_\gamma \times \phi_\gamma))_* [(z,z^\prime)] $$ and we now apply the lemma.