Let $R$ be a commutative ring and $(\varphi_i: (M_i \to N_i))_{i \in I}$ a family of $R$-module morphisms. Take
$$\varphi= \bigoplus \varphi: \bigoplus_i M_i \to \bigoplus_i N_i$$
Questions:
Does the cokernel $coker(\varphi)$ commute with direct sum? That is do we have always $coker(\bigoplus_i \varphi)= \bigoplus_i coker(\varphi_i)$?
If we now deal instead of direct sums more generally with direct limits and maps
$$\phi := \varinjlim_i \phi_i: \varinjlim_i M_i \to \varinjlim_i N_i $$
between direct limits of direct systems $(M_i)_i$ and $(N_i)_i$ and compatible maps $\phi: M_i \to N_i$, does cokernel also commute with direct limit? That is do we have $coker(\varinjlim_i \phi)= \varinjlim_i coker(\phi_i)$.
Is the same true if we deal now with kernel instead of cokernel?
I think the first two questions should be true and can be proved by universal property of direct sum: for any $R$-module $A$ a morphism $f:\bigoplus_i N_i \to A$ factorize over $coker(\bigoplus_i \varphi)$ iff $f \circ \varphi $ is zero. But $f \circ \varphi $ is zero iff for all $i$ the compositions $f \circ \varphi_i $ are zeoes iff for all $i$ $f$ factorize over $coker(\varphi_i)$.
Universal property of direct sum says that a morphism $\bigoplus_i coker(\varphi_i) \to A$ is the same as a family of morphisms $coker(\varphi_i) \to A$, and therefore $\bigoplus_i coker(\varphi_i)$ and $coker(\varphi)$ share same universal property.
Is my argument fine? Is there a another more simple one known? If it wirks then it should be not hart the extend it to direct limit uning now the universal property of directs limits instead, okay?
What about the case if we talk about kernel of $\varphi$ instead of cokernel?