$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \circ \Hom(\varepsilon_C,M)= id_M$. My primary references for contramodules are Positselski - Contramodules and Positselski - Contramodules over pro-perfect topological rings.

The category of $C$-contramodules is cocomplete (see, for instance, §1.7, Positselski - Contramodules over pro-perfect topological rings). However, colimit in the $C$-contramodule category doesn't appear to be the one coming from the underlying vector space structure. It is not clear to me how the colimit in this category is defined. Any help will be great! Further, I want to understand if a contramodule over $C$ is the direct limit of its finitely generated subcontramodules. We know that similar statement holds for a module over a ring and for a comodule over a coalgebra over a field. But unlike the category of modules and the category of comodules (over a coalgebra over a field), the $C$-contramodule category is not locally finitely generated.

$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \circ \Hom(\varepsilon_C,M)= id_M$ and $\pi_M \circ Hom(\Delta_C,M)=\pi \circ Hom(C,\pi)$ where $\Delta_C$ denotes the comultiplication in $C$. My primary references for contramodules are Positselski - Contramodules and Positselski - Contramodules over pro-perfect topological rings.

The category of $C$-contramodules is cocomplete (see, for instance, §1.7, Positselski - Contramodules over pro-perfect topological rings). However, colimit in the $C$-contramodule category doesn't appear to be the one coming from the underlying vector space structure. It is not clear to me how the colimit in this category is defined. Any help will be great! Further, I want to understand if a contramodule over $C$ is the direct limit of its finitely generated subcontramodules. We know that similar statement holds for a module over a ring and for a comodule over a coalgebra over a field. But unlike the category of modules and the category of comodules (over a coalgebra over a field), the $C$-contramodule category is not locally finitely generated.

Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:Hom_K(C,M)\longrightarrow M$ such that $\pi_M \circ Hom(\varepsilon_C,M)= id_M$. My primary references for contramodules are https://arxiv.org/abs/1503.00991 and https://arxiv.org/pdf/1807.10671.pdf.

The category of $C$-contramodules is cocomplete (see, for instance, $\S$ 1.7, https://arxiv.org/pdf/1807.10671.pdf). However, colimit in the $C$-contramodule category doesn't appear to be the one coming from the underlying vector space structure. It is not clear to me how the colimit in this category is defined. Any help will be great! Further, I want to understand if a contramodule over $C$ is the direct limit of its finitely generated subcontramodules. We know that similar statement holds for a module over a ring and for a comodule over a coalgebra over a field. But unlike the category of modules and the category of comodule (over a coalgebra over a field), the $C$-contramodule category is not locally finitely generated.

$\DeclareMathOperator\Hom{Hom}$Let $K$ be a field. Let $C$ be a $K$-coalgebra. A contramodule $M$ over $C$ is a $K$-space with a $K$-linear map $\pi_M:\Hom_K(C,M)\longrightarrow M$ such that $\pi_M \circ \Hom(\varepsilon_C,M)= id_M$. My primary references for contramodules are Positselski - Contramodules and Positselski - Contramodules over pro-perfect topological rings.

The category of $C$-contramodules is cocomplete (see, for instance, §1.7, Positselski - Contramodules over pro-perfect topological rings). However, colimit in the $C$-contramodule category doesn't appear to be the one coming from the underlying vector space structure. It is not clear to me how the colimit in this category is defined. Any help will be great! Further, I want to understand if a contramodule over $C$ is the direct limit of its finitely generated subcontramodules. We know that similar statement holds for a module over a ring and for a comodule over a coalgebra over a field. But unlike the category of modules and the category of comodules (over a coalgebra over a field), the $C$-contramodule category is not locally finitely generated.