Prove that a functor is additive Let $\mathcal{F}: Mod_A \rightarrow Mod_B$ be a covariant functor such that for all A-modules $M_1$ and $M_2$, the homomorphism $\mathcal{F}(M_1) \oplus \mathcal{F}(M_2) \rightarrow \mathcal{F}(M_1 \oplus M_2)$ that is induced by the canonical injection ${\mathcal{F}(i_k): \mathcal{F}(M_i) \rightarrow \mathcal{F}(M_1 \oplus M_2)}_{k=1,2}$ is an isomorphism, where $i_k: M_k \rightarrow M_1 \oplus M_2$. Show that $\mathcal{F}$ is additive. 
Being honest, I am a bit confused and, as I noticed, I must show that $Hom_A(M_1,M_2) \rightarrow Hom_B(\mathcal{F}(M_1), \mathcal{F}(M_2))$ is a group homomorphism, but I do now know how. Could anyone help?
Think a little bit more, I could tipe this solution, but I am not certain of it:
In order to show that $\mathcal{F}$ is additive, we must show that the map $\Psi: Hom_A(M_1,M_2) \rightarrow Hom_B(\mathcal{F}(M_1),\mathcal{F}(M_2)$ is a group homomorphism, i.e, $\Psi(f+g) = \Psi(f) \oplus \Psi(g)$. First of all, consider the map $f: M_1 \rightarrow M_2$ such that
\begin{tikzpicture}
\node (A) at (0,2) {$M_1$};
\node (B) at (2,2) {$M_1 \oplus M_1$};
\node (C) at (5,2) {$M_2 \oplus M_2$};
\node (D) at (8,2) {$M_2$};
\node (E) at (0.8,2.2) {$\sigma_f$};
\node (F) at (3.5,2.2) {$\Delta_f$};
\node (G) at (6.8,2.2) {$\alpha_f$};
\draw[->] (A) to (B);
\draw[->] (B) to (C);
\draw[->] (C) to (D);
\end{tikzpicture}
That is mapping an element $m_1 \rightarrow (m_1,m_1) \rightarrow (\Delta_f(m_1),\Delta_f(m_1)) \rightarrow \Delta_f(m_1)$, or, in other words, $f(m_1) = \Delta_f(m_1)$. And so the same for $g: M_1 \rightarrow M_2$. Also, consider the map $\Psi(f): \mathcal{F}(M_1) \rightarrow \mathcal{F}(M_2)$ such that
\begin{tikzpicture}
\node (A) at (0,2) {$\mathcal{F}(M_1)$};
\node (B) at (4,2) {$\mathcal{F}(M_1) \oplus \mathcal{F}(M_1)$};
\node (C) at (8.5,2) {$\mathcal{F}(M_2) \oplus \mathcal{F}(M_2)$};
\node (D) at (12,2) {$\mathcal{F}(M_2)$};
\node (E) at (1.5,2.2) {$\sigma_{\Psi(f)}$};
\node (F) at (6.2,2.2) {$\Delta_{\Psi(f)}$};
\node (G) at (10.5,2.2) {$\alpha_{\Psi(f)}$};
\draw[->] (A) to (B);
\draw[->] (B) to (C);
\draw[->] (C) to (D);
\end{tikzpicture}
That is mapping an element $\mathcal{F}(m_1) \rightarrow (\mathcal{F}(m_1),\mathcal{F}(m_1)) \rightarrow (\Delta_{\Psi(f)}(\mathcal{F}(m_1),\Delta_{\Psi(f)}(\mathcal{F}(m_1)) =( \mathcal{F}(\Delta_f(m_1)),\mathcal{F}(\Delta_f(m_1))) \rightarrow \Delta_{\Psi(f)} (\mathcal{F}(m_1))$, or, in other words, $\Psi(f)(m_1) = \Delta_{\Psi(f)}(\mathcal{F}(m_1))$ and the same for $\Psi(g)$. Now, see that $\Psi(f+g)_{\mathcal{F}(m_1)} = \mathcal{F}(\Delta_f(m_1) + \Delta_g(m_1)) = \mathcal{F}(\Delta_f(m_1)) \oplus \mathcal{F}(\Delta_g(m_1)) = \Delta_{\Psi(f)}(\mathcal{F}(m_1)) \oplus \Delta_{\Psi(g)}(\mathcal{F}(m_1)) = \Psi(f)_{\mathcal{F}(m_1)} \oplus \Psi(g)_{\mathcal{F}(m_1)}$ and, so, $\Psi$ is a group homomorphism and $\mathcal{F}$ is additive.
 A: In order to show that $\mathcal{F}$ is additive, we must show that the map $\Psi: Hom_A(M_1,M_2) \rightarrow Hom_B(\mathcal{F}(M_1),\mathcal{F}(M_2)$ is a group homomorphism, i.e, $\Psi(f+g) = \Psi(f) \oplus \Psi(g)$. First of all, consider the map $f: M_1 \rightarrow M_2$ such that
\begin{tikzpicture}
\node (A) at (0,2) {$M_1$};
\node (B) at (2,2) {$M_1 \oplus M_1$};
\node (C) at (5,2) {$M_2 \oplus M_2$};
\node (D) at (8,2) {$M_2$};
\node (E) at (0.8,2.2) {$\sigma_f$};
\node (F) at (3.5,2.2) {$\Delta_f$};
\node (G) at (6.8,2.2) {$\alpha_f$};
\draw[->] (A) to (B);
\draw[->] (B) to (C);
\draw[->] (C) to (D);
\end{tikzpicture}
That is mapping an element $m_1 \rightarrow (m_1,m_1) \rightarrow (\Delta_f(m_1),\Delta_f(m_1)) \rightarrow \Delta_f(m_1)$, or, in other words, $f(m_1) = \Delta_f(m_1)$. And so the same for $g: M_1 \rightarrow M_2$. Also, consider the map $\Psi(f): \mathcal{F}(M_1) \rightarrow \mathcal{F}(M_2)$ such that
\begin{tikzpicture}
\node (A) at (0,2) {$\mathcal{F}(M_1)$};
\node (B) at (4,2) {$\mathcal{F}(M_1) \oplus \mathcal{F}(M_1)$};
\node (C) at (8.5,2) {$\mathcal{F}(M_2) \oplus \mathcal{F}(M_2)$};
\node (D) at (12,2) {$\mathcal{F}(M_2)$};
\node (E) at (1.5,2.2) {$\sigma_{\Psi(f)}$};
\node (F) at (6.2,2.2) {$\Delta_{\Psi(f)}$};
\node (G) at (10.5,2.2) {$\alpha_{\Psi(f)}$};
\draw[->] (A) to (B);
\draw[->] (B) to (C);
\draw[->] (C) to (D);
\end{tikzpicture}
That is mapping an element $\mathcal{F}(m_1) \rightarrow (\mathcal{F}(m_1),\mathcal{F}(m_1)) \rightarrow (\Delta_{\Psi(f)}(\mathcal{F}(m_1),\Delta_{\Psi(f)}(\mathcal{F}(m_1)) =( \mathcal{F}(\Delta_f(m_1)),\mathcal{F}(\Delta_f(m_1))) \rightarrow \Delta_{\Psi(f)} (\mathcal{F}(m_1))$, or, in other words, $\Psi(f)(m_1) = \Delta_{\Psi(f)}(\mathcal{F}(m_1))$ and the same for $\Psi(g)$. Now, see that $\Psi(f+g)_{\mathcal{F}(m_1)} = \mathcal{F}(\Delta_f(m_1) + \Delta_g(m_1)) = \mathcal{F}(\Delta_f(m_1)) \oplus \mathcal{F}(\Delta_g(m_1)) = \Delta_{\Psi(f)}(\mathcal{F}(m_1)) \oplus \Delta_{\Psi(g)}(\mathcal{F}(m_1)) = \Psi(f)_{\mathcal{F}(m_1)} \oplus \Psi(g)_{\mathcal{F}(m_1)}$ and, so, $\Psi$ is a group homomorphism and $\mathcal{F}$ is additive.
