The understanding of category of groups

Question

When we say the category of groups, do we refer to the same category of groups? For example, in Tom's mind, there is a category of groups; in Jack's mind, there is a category of groups. However, one day Tom constructs a group $A$ nobody else knows, thought $A$ is isomorphic to some group $B$ contained in Jack's category of groups, can we regard $A$ contained in Jack's category of groups? When we say a set of people, we mean the same set of people, though we don't know everybody who lives on the earth, but the existence of everybody is objective. But the existence of group $A$ is not objective, it only exists in Tom's mind. Then how do we correctly understand the category of groups?