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This paper emerged as a result of tackling the following three issues.Firstly, we would like the well known embedding of bicategories into pseudodouble categories to be monoidal, which it is not if one uses the usual notionof a monoidal pseudo double category. Secondly, in \cite{Gabi} the question wasraised: which would be an alternative notion to intercategories of Grandis andPar\'e, so that monoids in B\"ohm's monoidal category $(Dbl,\ot)$ of strictdouble categories and strict double functors with a Gray type monoidal productbe an example of it? We obtain and prove that precisely the monoidal structureof $(Dbl,\ot)$ resolves the first question. On the other hand, resolving thesecond question, we upgrade the category $Dbl$ to a tricategory $\DblPs$ andpropose %an alternative definition of intercategories as to consider internalcategories in this tricategory. %, enabling monoids in $(Dbl,\ot)$ to beexamples of this gadget. Apart from monoids in $(Dbl,\ot)$ - more importanlty,weak pseudomonoids in a tricategory containing $(Dbl,\ot)$ as a sub 1-category- most of the examples of intercategories are also examples of this gadget, theones that escape are those that rely on laxness of the product on the pullback,as duoidal categories. For the latter purpose we define categories internal totricategories (of the type of $\DblPs$), which simultaneously serves our thirdmotive. Namely, inspired by the tricategory and $(1\times 2)$-category oftensor categories, we prove under mild conditions that categories enriched overcertain type of tricategories may be made into categories internal in them. Weillustrate this occurrence for tensor categories with respect to the ambienttricategory $2\x\Cat_{wk}$ of 2-categories, pseudofunctors, pseudonaturaltransformations and modifications.

Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories