What is a relation of category of complete sublattices to category of lattices? What is a relation of category of complete sublattices to category of lattices?
I haven´t found much about a category of lattices, but I assume objects = lattices, morphisms = lattice homomorphisms.
My question is, do also complete sublattices of a given lattice form a category? And do complete sublattices have any special relation to the category of lattices?
I am just interested in whether all subsets having supremum and infimum can "do something" in a categorial language.
 A: I have talked about my question with professors at my university, so I will summarize their advice and post my own answer.
Lattice as a category
First, from a categorial view, any lattice is a poset $P$, where objects are elements of $P$ and $Hom(x,y)$ has just one element if $x \leq y$ and is empty otherwise. Composition is of course well defined, it is provided by the transitivity of the partial order.
If $x, y \in P$, then the coproduct of $x$ and $y$ is the same as the supremum of $x$ and $y$ in $P$ and products are the same as infima. So P is a lattice iff the corresponding category has finite products and coproducts.
$P$ defined this way will also be a small category.
Complete lattice as a category
$P$ is a complete lattice iff the category has all products and coproducts.
Full subcategory of the category $P$
If $P$ is a poset (and a category, defined as above), then take a sub-poset $Q$ with ordering inherited from $Q$. If $Q$ is closed under suprema and infima of all pairs $x, y \in P$, then $Q$ is a sublattice of $P$.
In categorial language, this is the same as saying that $Q$ is closed under $Q$ is closed under all products and co-products of pairs in $P$. In this case, $Q$ is automatically a full sub-category of $P$.
Other (not full) subcategories of a category $P$

*

*Semilattice S and its sub-semilattice Q

Q is a sub-category of P iff it is closed under all products of pairs (if P is a join-semilattice), or all co-products of pairs (if P is a meet-semilattice).


*Complete sublattice of P

If Q is complete sublattice of P, it is subcategory of P iff it is closed under products and coproducts of all finite subsets of Q.
