The answer is yes, one can lose categoricity in a reduct of a theory. Consider the following example.

Consider the theory $T$ describing a bijection between two disjoint infinite predicates $f:A\to B$. So a model consists of two disjoint parts, the $A$-part and the $B$-part, and a bijection $f$ between them. The language is $\{f,A,B\}$.

This theory is categorical in every cardinality. But if we restrict the theory to its consequences in the language with the two predicates $\{A,B\}$ and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in uncountable powers, since one predicate could have a different cardinality than the other.

The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.

Consider the theory $T$ describing a bijection between two disjoint infinite predicates $f:A\to B$. So a model consists of two disjoint parts, the $A$-part and the $B$-part, and a bijection $f$ between them. The language is $\{f,A,B\}$.

This theory is categorical in every cardinality. But if we restrict the theory to its consequences in the language with the two predicates $\{A,B\}$ and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in uncountable powers, since one predicate could have a different cardinality than the other.

The answer is yes, this can happen. Consider the following example.

Consider the theory describing a bijection between two disjoint predicates $f:A\to B$. So a model consists of two disjoint parts, the $A$-part and the $B$-part, and a bijection $f$ between them. The language is $\{f,A,B\}$.

This theory is categorical in every cardinality. But if we restrict the theory to just the language with the two predicates $\{A,B\}$ and without the bijection, then it is no longer categorical in uncountable powers, since one predicate could have a different cardinality than the other.

The answer is yes, one can lose categoricity in a reduct of a theory. Consider the following example.

Consider the theory $T$ describing a bijection between two disjoint infinite predicates $f:A\to B$. So a model consists of two disjoint parts, the $A$-part and the $B$-part, and a bijection $f$ between them. The language is $\{f,A,B\}$.

This theory is categorical in every cardinality. But if we restrict the theory to its consequences in the language with the two predicates $\{A,B\}$ and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in uncountable powers, since one predicate could have a different cardinality than the other.