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Then you can show that there is an isomorphism of $\mathbb T_{\mathfrak m}[G_{\mathbb Q}\times GL_2(\mathbb Q_p) \times \prod_{i = 1}^n GL_2(\mathbb Q_{\ell_i}]$-modules

Then you can show that there is an isomorphism of $\mathbb T_{\mathfrak m}[G_{\mathbb Q}\times GL_2(\mathbb Q_p) \times \prod_{i = 1}^n GL_2(\mathbb Q_{\ell_i})]$-modules

Added: You asked about motivation. The original motivation for defining completed cohomology was to construct eigenvarieties. Later it became clear that it was an important object in its own right, providing a global counterpart to the representations of $p$-adic groups that were beginning to appear as part of $p$-adic local Langlands. E.g. the theorem that locally algebraic vectors in cohomology are classical was first proved as an ingredient in the proof of an analogue of Coleman's "small slope implies classical" result for the eigenvariety constructed from completed cohomology. Only later was it realized that this could be combined with a local-global compatiblity result to prove modularity theorems for Galois representations.

Note that the rough relation with $p$-adic modular forms, namely that one gets the same Hecke algebra via either approach, was clear from the beginning, even though the more precise Eichler--Shimura-like statement above was not. Since eigenvarieties (as their name indicates) only care about Hecke eigenvalues, this meant that completed cohomology was good enough for constructing them.

It brings out the difference between the two sides quite clearly: on the completed cohomology side, we have a Galois action, which is encoded in the appearance of $\rho^u$. (Classically, this reflects the fact that every cuspform appears "twice" in cohomology.)

Note also that this isomorphism is not canonical. In this sense, it is analogous to looking at classical cohomology of modular forms with say $\mathbb Q$-coefficients, and modular forms with $\mathbb Q$-coefficients. These will be isomorphic as Hecke modules --- up to the issue of cuspforms appearing twice in cohomology --- but not canonically so. In order to make the Eichler--Shimura isomorphism canonical, one has to extend scalars to an appropriate period ring. Whether this is possible with completed cohomology I'm not sure about at the moment.

It brings out the difference between the two sides quite clearly: on the completed cohomology side, we have a Galois action, which is encoded in the appearance of $\rho^u$. (This reflects the classical fact that every cuspform appears "twice" in cohomology.)

Note also that this isomorphism is not canonical. In this sense, it is analogous to looking at classical cohomology of modular forms with say $\mathbb Q$-coefficients, and modular forms with $\mathbb Q$-coefficients. These will be isomorphic as Hecke modules --- up to the issue of cuspforms appearing twice in cohomology --- but not canonically so. In order to make the Eichler--Shimura isomorphism canonical, one has to extend scalars to an appropriate period ring. Whether this is possible with completed cohomology I'm not sure about at the moment.

One more remark: trading in a $U_p$-action for a $GL_2(\mathbb Q_p)$-action is a fairly significant upgrading of structure, and this is why completed cohomology provides a useful tool for proving modularity theorems for Galois representations, over and above the already-existing theories of $p$-adic modular forms and $p$-adic modular symbols.