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The category of left exact functors $E\colon\mathcal A_p^{op}\to\mathcal Ab$ is equivalent to the category of $p$-primary torsion abelian groups $T$. To a $p$-primary torsion abelian group $T$ one assigns the contravariant functor $E_T$ taking a finite abelian $p$-group $A$ to the abelian group $\operatorname{Hom}_{\mathbb Z}(A,T)$. To construct the inverse functor, consider the projective system of finite abelian groups $\mathbb Z/p\mathbb Z \leftarrow \mathbb Z/p^2\mathbb Z\leftarrow \mathbb Z/p^3\mathbb Z\leftarrow\dotsb$ and apply the contravariant functor $E$ to it. The $p$-primary torsion abelian group $T_E$ corresponding to $E$ is the inductive limit of the sequence $E(\mathbb Z/p\mathbb Z)\to E(\mathbb Z/p^2\mathbb Z)\to E(\mathbb Z/p^3\mathbb Z)\to\dotsb$. From the left exactness property of the functor $E$ one can see that the group $E_m=E(\mathbb Z/p^m\mathbb Z$) is identified with the subgroup of elements annihilated by $p^m$ in the group $E_n$, for every $n\ge m$.

The counterexample showing that the categories of $p$-separated $p$-complete abelian groups and $x$-separated $x$-complete $k[x]$-modules are not abelian is now well-known. It has been rediscovered and discussed by many authors, including Example 2.5 in A.-M. Simon, "Approximations of complete modules by complete big Cohen-Macaulay modules over a Cohen-Macaulay local ring", Algebras and Represent. Theory 12, 2009, and Example 3.20 in A. Yekutieli, "On flatness and completion in infinitely generated modules over Noetherian rings", Communic. in Algebra 39, 2010, https://arxiv.org/abs/0902.4378 .

The category of left exact functors $E\colon\mathcal A_p^{op}\to\mathcal Ab$ is equivalent to the category of $p$-primary torsion abelian groups $T$. To a $p$-primary torsion abelian group $T$ one assigns the contravariant functor $E_T$ taking a finite abelian $p$-group $A$ to the abelian group $\operatorname{Hom}_{\mathbb Z}(A,T)$. To construct the inverse functor, consider the projective system of finite abelian $p$-groups $\mathbb Z/p\mathbb Z \leftarrow \mathbb Z/p^2\mathbb Z\leftarrow \mathbb Z/p^3\mathbb Z\leftarrow\dotsb$ and apply the contravariant functor $E$ to it. The $p$-primary torsion abelian group $T_E$ corresponding to $E$ is the inductive limit of the sequence $E(\mathbb Z/p\mathbb Z)\to E(\mathbb Z/p^2\mathbb Z)\to E(\mathbb Z/p^3\mathbb Z)\to\dotsb$. From the left exactness property of the functor $E$ one can see that the group $E_m=E(\mathbb Z/p^m\mathbb Z$) is identified with the subgroup of elements annihilated by $p^m$ in the group $E_n$, for every $n\ge m$.

The categories of $p$-separated $p$-complete abelian groups and $x$-separated $x$-complete $k[x]$-modules are complete and cocomplete. In fact, they are locally $\aleph_1$-presentable, being reflective and closed under $\aleph_1$-filtered colimits as full subcategories in the locally $\aleph_1$-presentable abelian categories of Ext-$p$-complete abelian groups and Ext-$x$-complete $k[x]$-modules (respectively). So, in particular, all morphisms in these two categories (of right exact functors) have kernels and cokernels. Still, these two categories are not abelian.

The counterexample showing that the categories of $p$-separated $p$-complete abelian groups and $x$-separated $x$-complete $k[x]$-modules are not abelian is now well-known. It has been rediscovered and discussed by many authors, including Example 2.5 in A.-M. Simon, "Approximations of complete modules by complete big Cohen-Macaulay modules over a Cohen-Macaulay local ring", Algebras and Represent. Theory 12, 2009, and Example 3.20 in A. Yekutieli, "On flatness and completion for infinitely generated modules over noetherian rings", Communic. in Algebra 39, 2010, https://arxiv.org/abs/0902.4378 .

Let $p$ be a prime number, and let $\mathcal A_p$ be the abelian category of finite abelian $p$-groups. Let $F\colon\mathcal A_p\to\mathcal Ab$ be a right exact functor. Then $F_n=F(\mathbb Z/p^n\mathbb Z)$ is an abelian group annihilated by the multiplication with $p^n$. For any $n\ge m$, from the right exact sequence $\mathbb Z/p^n\mathbb Z\to \mathbb Z/p^n\mathbb Z \to \mathbb Z/p^m\mathbb Z\to 0$ we get an isomorphism $F_n/p^mF_n\cong F_m$. The functor $F$ is uniquely determined by the sequence of abelian groups $F_n$ together with these isomorphisms. Indeed, for every abelian group $A$ annihilated by $p^n$ one has $F(A)=F_n\otimes_{\mathbb Z/p^n\mathbb Z} A$.

Let $p$ be a prime number, and let $\mathcal A_p$ be the abelian category of finite abelian $p$-groups. Let me start with a very brief introductory discussion of the category of left exact functors $\mathcal A_p\to\mathcal Ab$. The category of finite abelian $p$-groups is self-dual (by Pontryagin duality, taking a finite abelian $p$-group $A$ to the group $\operatorname{Hom}_{\mathbb Z}(A,\mathbb Q/\mathbb Z)$). So the category of left exact functors $\mathcal A_p\to\mathcal Ab$ is equivalent to the category of left exact functors $\mathcal A_p^{op}\to\mathcal Ab$.

The category of left exact functors $E\colon\mathcal A_p^{op}\to\mathcal Ab$ is equivalent to the category of $p$-primary torsion abelian groups $T$. To a $p$-primary torsion abelian group $T$ one assigns the contravariant functor $E_T$ taking a finite abelian $p$-group $A$ to the abelian group $\operatorname{Hom}_{\mathbb Z}(A,T)$. To construct the inverse functor, consider the projective system of finite abelian groups $\mathbb Z/p\mathbb Z \leftarrow \mathbb Z/p^2\mathbb Z\leftarrow \mathbb Z/p^3\mathbb Z\leftarrow\dotsb$ and apply the contravariant functor $E$ to it. The $p$-primary torsion abelian group $T_E$ corresponding to $E$ is the inductive limit of the sequence $E(\mathbb Z/p\mathbb Z)\to E(\mathbb Z/p^2\mathbb Z)\to E(\mathbb Z/p^3\mathbb Z)\to\dotsb$. From the left exactness property of the functor $E$ one can see that the group $E_m=E(\mathbb Z/p^m\mathbb Z$) is identified with the subgroup of elements annihilated by $p^m$ in the group $E_n$, for every $n\ge m$.

The category of $p$-primary torsion abelian groups is abelian. Hence so is the category of left exact functors $\mathcal A_p\to \mathcal Ab$ (as we know it should be by the additive sheaf theory).

Now let us turn to right exact functors $\mathcal A_p\to \mathcal Ab$, which we are really interested in. Let $F\colon\mathcal A_p\to\mathcal Ab$ be a right exact functor. Then $F_n=F(\mathbb Z/p^n\mathbb Z)$ is an abelian group annihilated by the multiplication with $p^n$. For any $n\ge m$, from the right exact sequence $\mathbb Z/p^n\mathbb Z\to \mathbb Z/p^n\mathbb Z \to \mathbb Z/p^m\mathbb Z\to 0$ we get an isomorphism $F_n/p^mF_n\cong F_m$. The functor $F$ is uniquely determined by the sequence of abelian groups $F_n$ together with these isomorphisms. Indeed, for every abelian group $A$ annihilated by $p^n$ one has $F(A)=F_n\otimes_{\mathbb Z/p^n\mathbb Z} A$.

Similarly, let $k$ be a field, and let $\mathcal A_k$ be the abelian category of finite-dimensional $k$-vector spaces $V$ endowed with a nilpotent $k$-linear operator $x\colon V\to V$. Let $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$ be a right exact functor. Then $G_n=G(k[x]/x^nk[x])$ is a $k[x]/x^nk[x]$-module. Just as in the first example above, one constructs a natural isomorphism $G_n/x^mG_n\cong G_m$ for every $n\ge m$. The functor $G$ is uniquely determined by the sequence of modules $G_n$ together with these isomorphisms.

To any right exact functor $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$, one assigns the $k[x]$-module (or, if one wishes, $k[[x]]$-module) $D_G=\varprojlim_n G_n$. This assignment is an equivalence between the category of right exact functors $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$ and the full subcategory in $k[x]$-modules (or, equivalently, in $k[[x]]$-modules) consisting of all the $x$-separated and $x$-complete modules, i.e., $k[x]$-modules or $k[[x]]$-modules $D$ such that the natural map $D\to\varprojlim_n D/x^nD$ is an isomorphism. The inverse functor assigns to a $k[x]$-module $D$ the functor $G_D$ taking a $k$-finite-dimensional $k[x]$-module $B$ with $x$ acting by a nilpotent operator to the $k$-vector space $D\otimes_{k[x]}B$. The corresponding modules $G_n$ are $G_n=D/x^nD$.

Similarly, let $k$ be a field, and let $\mathcal A_k$ be the abelian category of finite-dimensional $k$-vector spaces $V$ endowed with a nilpotent $k$-linear operator $x\colon V\to V$. Let $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$ be a $k$-linear right exact functor. Then $G_n=G(k[x]/x^nk[x])$ is a $k[x]/x^nk[x]$-module. Just as in the first example above, one constructs a natural isomorphism $G_n/x^mG_n\cong G_m$ for every $n\ge m$. The functor $G$ is uniquely determined by the sequence of modules $G_n$ together with these isomorphisms.

To any $k$-linear right exact functor $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$, one assigns the $k[x]$-module (or, if one wishes, $k[[x]]$-module) $D_G=\varprojlim_n G_n$. This assignment is an equivalence between the category of $k$-linear right exact functors $G\colon\mathcal A_k\to k{-}\mathcal{V}ect$ and the full subcategory in $k[x]$-modules (or, equivalently, in $k[[x]]$-modules) consisting of all the $x$-separated and $x$-complete modules, i.e., $k[x]$-modules or $k[[x]]$-modules $D$ such that the natural map $D\to\varprojlim_n D/x^nD$ is an isomorphism. The inverse functor assigns to a $k[x]$-module $D$ the functor $G_D$ taking a $k$-finite-dimensional $k[x]$-module $B$ with $x$ acting by a nilpotent operator to the $k$-vector space $D\otimes_{k[x]}B$. The corresponding modules $G_n$ are $G_n=D/x^nD$.

Taking $k=\mathbb Z/p\mathbb Z$ or $k=\mathbb Q$, one can drop the adjective "$k$-linear" before the words "right exact functor" (as any abelian group homomorphism between $k$-vector spaces is a $k$-linear map, so any additive functor between $k$-linear categories is $k$-linear, for such fields $k$).