Properties preserved under equivalence of categories I would like to ask about properties that are preserved under equivalence of categories. To be more specific, is it true that equivalences preserve limits? Why?
 A: Basically any property that can be considered categorical in nature. Any textbook would list a warning if a property isn't preserved. Wikipedia lists some simple examples.
Here are some things that aren't necessarily preserved:

*

*Number of objects

*Number of morphisms (total)

*Underlying graph

*Other evil properties

Tip for Proofs:
Equivalences preserve hom-sets. This helps, for example, if you are trying to proof that a morphism is unique.
A: In Categories, Allegories, Freyd and Scedrov describe a diagrammatic form of logic. The simplest kind of proposition consists of a finite sequence of categories
$$ J_0 \xrightarrow{i_0} J_1 \xrightarrow{i_1} \ldots \xrightarrow{i_{n-1}} J_n $$
A functor $F_0 : J_0 \to C$ satisfies this statement if and only if:


*

*there exists a functor $F_1 : J_1 \to C$ such that $F_1 i_0 = F_0$ such that

*for every functor $F_2 : J_2 \to C$ such that $F_2 i_1 = F_1$ such that

*there exists a functor $F_3 : J_3 \to C$ such that $F_3 i_2 = F_2$ such that

*...


For example, the proposition that a $J$-shaped diagram has a limit is given by the sequence (all of the $i$ maps are inclusions)


*

*$J_0 = J$

*$J_1$ is the diagram that includes the object, arrows, and equations describing a cone 

*$J_2$ is the diagram that includes the object, arrows, and equations describing another cone 

*$J_3$ is the diagram that includes the arrow and equations describing a morphism between the cones

*$J_4$ is the diagram that includes the arrow and equations describing another morphism between the cones

*$J_5$ includes the relations asserting both of those arrows are equal


The theorem they prove is

A proposition respects equivalence of categories if and only if it can be described on a blackboard

which is an amusing description whose actual content is

A proposition respects equivalence of categories if and only if every $i_n$ is injective on objects

The meat of the theorem is part of the canonical model structure:

A diagram of categories and functors
  $$ \require{AMScd} \begin{CD}
A @>F>> B
\\ @VGVV @VHVV
\\ C @>K>> D
\end{CD} $$
  has the lifting property iff there exists a functor $L:C \to B$ that makes the diagram commute: that is, $F = LC$ and $K = HL$.
Theorem: If there is a specific functor $G$ such that every square where $H$ is a surjective equivalence has the lifting property, then $G$ is injective on objects

To see why this is relevant, suppose we have
$$ \require{AMScd} \begin{CD}
J_n @>F_n>> C
\\ @Vi_nVV @VVUV
\\ J_{n+1} & & D
\end{CD} $$
where $U$ is a surjective equivalence. Define $G_n = U F_n$ and suppose $i_n$ is injective on objects.
Regarding satisfying propositions, we ask about the possible existence of functors $F_{n+1} : J_{n+1} \to C$ and $G_{n+1} : J_{n+1} \to D$ that satisfy $F_{n} = F_{n+1} i_n$ and $G_n = G_{n+1} i_n$, and are still related by $G_{n+1} = U F_{n+1}$.


*

*If we're given an $F_{n+1}$, then it's clear that we can also construct $G_{n+1}$ simply by setting $G_{n+1} = U F_{n+1}$.

*If we're given a $G_{n+1}$, then it completes the square above, and the lifting property gives us an $F_{n+1}$.


This can be used to show that $F_0 : J_0 \to C$ satisfies a sentence if and only if $U F_0 : J_0 \to D$ satisfies the sentence.
