In Johnstone's Sketch of Elephant. Page 1034, section D5.1, Lemma 5.1.1 says:
Any $T$-algebra contains a unique smallest closed subobject, which is recursive.
In the proof of "Any $T$-algebra contains a unique smallest closed subobject". He says:
An arbitrary intersection of sub-$T$-algebra is a sub-$T$-algebra, so we may obtain the smallest one by forming the intersection of all of them.
My question is that this proof mentions "arbitrary intersection", which may not exist in a category that is not complete (even in a topos under Lawvere's ETCS definition, we can only take finite limit). Does this proof assume that we are working in a complete category? If it does not, how does it work if we do not assume arbitrary limit exists?
The lemma in question concerns the algebraic theory $\mathbb{T}$ freely generated by a constant and a unary operation. This is important because it means the structure of a $\mathbb{T}$-algebra can be expressed in the internal logic of elementary toposes. In particular, the property of a subobject of the underlying object of a $\mathbb{T}$-algebra being a sub-$\mathbb{T}$-algebra can be expressed in the internal logic. Therefore we may form (as a subobject of the powerobject) the object of all sub-$\mathbb{T}$-algebras. Being indexed by a subobject of the powerobject, we can therefore take the intersection of all sub-$\mathbb{T}$-algebras of the given $\mathbb{T}$-algebra. All this happens in the internal logic, so there is no need to invoke any external completeness.
Moral: if something looks like it needs infinite limits or colimits but you need it to work in an elementary topos, try using internal logic instead.