I'd like to know if certain properties regarding preservation and reflection of (co)limits hold for the functor $w$ defined below. The question is a bit long, but you can just take a look at the definitions, and skip to the question; the preliminary facts are there to justify the claims and to understand where eventual errors could come from. Thank you.
Definitions:
- a short sequence in an abelian category $\sf A$ is a diagram of shape $0\to a'\to a\to a''\to 0$, such that the composition of the central morphisms is $0$;
- let $w:\mathsf A\to [\mathsf A^{op}, \mathsf{Ab}]$ be the functor defined by $a\mapsto \mathsf A(-,a)$;
- let $u:\sf Ab\to Set$ be the forgetful functor, and define the functor $v: [\mathsf A^{op}, \mathsf{Ab}]\to [\mathsf A^{op}, \mathsf{Set}]$ by $v(h)=uh$;
- set $y:=vw$ the Yoneda embedding.
Preliminary facts:
(A) $y$ preserves and reflects limits.
(B) Take three functors $f,g,l$ such that $fg=l$. If $l$ and $f$ preserve and reflect a certain class of (co)limits, $g$ preserves and reflects it too.
(C) Let $\sf C$, $\sf D$ be any two categories. Viewing the class $\operatorname{ob}(\mathsf C)$ as a discrete category, let $i$ be the canonical functor $\operatorname{ob}(\mathsf C)\to \mathsf C$. Define the functor $r:[\mathsf C,\mathsf D]\to[\operatorname{ob}(\mathsf C),\mathsf D]$ by $r(h)=hi$. Now, $r$ preserves and reflects limits and colimits.
(D) The fact that $u$ preserves and reflects limits, together with (C), shows that $v$ preserves and reflects limits.
(E) From now on assume that $\sf D$ is abelian. Also $[\mathsf C,\mathsf D]$ is abelian, and a short sequence $0\to h'\to h\to h''\to 0$ in $[\mathsf C,\mathsf D]$ is exact iff the short sequences $0\to h'(c)\to h(c)\to h''(c)\to 0$ in $\sf D$ are exact for all $c$ in $\sf C$, applying (C).
(F) A short sequence in $\mathsf D$, say$0\to d'\to d\to d''\to 0$, is exact iff the following short sequence in $\sf Ab$ is exact for all $b\in \sf D$: $$0\to\mathsf D(b,d')\to\mathsf D(b,d)\to\mathsf D(b,d'')\to 0.$$
Question. Are the italic sentences below true?
(1) $w$ is additive, so preserves finite coproducts, and it preserves short exact sequences by (E) and (F). In particular it preserves cokernels, so $w$ preserves finite colimits.
(2) $w$ preserves and reflects limits by (A), (B) and (D).
(3) for the remaining properties, do you confirm that $w$ doesn't reflect colimits and $y$ doesn't even preserve them? (I know that the Yoneda embedding doesn't preserve colimits in general, but here we are assuming that $\sf A$ is abelian). I have not found counterexamples but it seems very reasonable that these properties are not satisfied.
Correction after the answers. (F) is false, only the "if" part holds (that's why I was confused about preservation of cokernels).
To start off with, let us recall that limits and colimits in a presheaf-of-abelian-groups category $[\mathsf{C}, \mathsf{Ab}]$ are computed "pointwise", i.e. for a natural transformation $\alpha : F \to G$, $\ker(\alpha) \simeq (a \mapsto \ker(\alpha_a))$, $\operatorname{cok}(\alpha) \simeq (a \mapsto \operatorname{cok}(\alpha_a))$, etc. Therefore, to ask whether $w$ preserves cokernels is equivalent to asking whether for each object $a$ of $C$, we have that $\mathsf{C}(a, {-})$ preserves cokernels (since we have a canonical comparison morphism of functors anyway of which we are asking whether it is an isomorphism, and a morphism of functors is an isomorphism if and only if it is pointwise an isomorphism).
With this in mind, let us go through the questions:
(1) $w$ preserves cokernels if and only if every object of $\mathsf{A}$ is projective -- which for example is true for the category of vector spaces over a field, but not true for $\mathsf{A} = \mathsf{Ab}$.
(2) $w$ indeed preserves and reflects limits.
(3) $w$ preserves a coproduct $\bigoplus_{i\in I} A_i$ if and only if for each object $a$ of $\mathsf{A}$, the comparison morphism $\bigoplus_{i\in I} \mathsf{A}(a, A_i) \to \mathsf{A}\left( a, \bigoplus_{i\in I} A_i \right)$ is an isomorphism. However, considering the case $a = \bigoplus_{i\in I} A_i$ and the identity morphism on the right, this would imply that the identity morphism is a finite sum of morphisms of the form $\iota_i \circ \varphi$ where $\varphi \in \mathsf{A}\left( \bigoplus_{i\in I} A_i, A_i \right)$ and $\iota_i \in \mathsf{A}\left( A_i, \bigoplus_{i\in I} A_i \right)$ is the canonical inclusion. That implies that $A_j$ is a zero object for any $j$ not in the finite set of indices $i$. On the other hand, if this condition holds, then $w$ is equivalent to a finite coproduct, which is preserved by the additive functor $w$. To summarize, we see that $w$ preserves a coproduct $\bigoplus_{i\in I} A_i$ if and only if all but finitely many $A_i$ are zero objects.
Incidentally, there is no reason to expect that $y$ preserves even finite coproducts. Since similarly to the above, coproducts in a presheaf category $[\mathsf{C}, \mathsf{Set}]$ are computed pointwise, $y$ preserves a finite coproduct $b \oplus c$ if and only if for every object $a$, every morphism in $\mathsf{A}(a, b \oplus c)$ is induced either by a morphism in $\mathsf{A}(a, b)$ or by a morphism in $\mathsf{A}(a, c)$, but not by both. The last condition is ridiculous for the case of the zero morphism of $\mathsf{A}(a, b \oplus c)$.