This paper provides a set of category-theoretic analyses of Gödel’s incompleteness theorems and Tarski’s indefinability theorem (see Appendix). We view the first-order theory as a mathematical language and introduce the notion of "language charge" as a monad within a category. For each analysis, we introduce a pair of adjunct categories: a syntactic category and a semantic category. We show that the Gödel numbering can be modeled as a pair of adjoint functors between these categories—a right functor from syntax to semantics and a left functor in the reverse direction. We prove that the Gödel numbering functor serves as a limit in a functor category. Additional analyses focus on the expressibility and definability in the twin theorems. Each of these is linked to natural transformations. In addition, we establish a formal account of "spontaneous naturality breaking" in the context of Gödel’s independent statements and Tarski’s indefinability. Finally, it touches higher order categories. By composing a syntactic category and a semantic category, we constructed a 2-Category with two layers of structures. Note that 2-category is one of the current research interests in category theory. Further, by decomposing and recomposing the syntactic category, we constructed a 3-Category.