The paper deals with notions of completely and totally distributive categories. In 1978, Street and Walters defined a locally small category \(\mathcal{K}\) to be totally cocomplete if its Yoneda functor \(Y\) has a left adjoint \(X\). Such a \(\mathcal{K}\) is \textit{totally distributive} if \(X\) has a left adjoint \(W\). A locally small category \(\mathcal{K}\) is small cocomplete if it is a \(\mathcal{P}\)-algebra, where \(\mathcal{P}\) is the small-colimit completion monad on \(\mathbf{Cat}\). \(\mathcal{K}\) is \textit{completely distributive} if \(\mathcal{K}\) is small cocomplete, small complete, and assignment of colimits \(X : \mathcal{PK}\rightarrow \mathcal{K}\) preserves small limits. The authors show that totally distributive categories are completely distributive and present examples of the former. Also, recall that a category \(\mathcal{K}\) is a (possibly large) set \(|\mathcal{K}|\) together with a monad \(\mathcal{K}\) on \(|\mathcal{K}|\) in \textbf{MAT}, the bicategory with objects those of \textbf{SET} and arrows given by \textbf{SET}-valued matrices. A \textit{taxon} \(\mathbf{T}\), as introduced by Koslowski in 1997, is a (possibly large) set \(|\mathbf{T}|,\) whose elements are called objects, together with an interpolad \(\mathbf{T}\) on \(|\mathbf{T}|\) in \textbf{MAT}. This means that \(\mathbf{T}\) is a pair \(\mathbf{T} = (\mathbf{T} : |\mathbf{T}| \nrightarrow |\mathbf{T}|, \mu : \mathbf{TT}\rightarrow \mathbf{T})\) in \textbf{MAT}, where \(\mu : \mathbf{TT}\rightarrow \mathbf{T}\) is a coequalizer in \(\mathbf{MAT}(|\mathbf{T}|, |\mathbf{T}|)\) of \(\mathbf{T}\mu\) and \(\mu\mathbf{T}.\) The authors show, for small taxons \(\mathbf{T}\) and \(\mathbf{S},\) that the category \(\mathbf i\)-\(\mathbf{mod}(\mathbf{T}, \mathbf{S})\) of \(i\)-modules between them is a totally distributive category.