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As hinted at before the universal Hecke algebra contains the Hecke operator $U_p$. We can then as usual define the ordinary projector as the idempotent $$e^{ord}=\lim\limits_{\xleftarrow{r}}D^{\vee}_{1,r}e^{ord}_t=\lim\limits_{\xleftarrow{r}}D^{\vee}_{1,r}\left(\lim\limits_{n\rightarrow\infty}U^{n!}_p\right).$$ Here $$D^{\vee}_{1,r}$$ denotes the dual of the degeneracy map defined above (which is a restriction to a specific set of cuspforms) and $e^{ord}_r$ denotes the ordinary projector of level $\Gamma_0(Np^r)$.

As hinted at before the universal Hecke algebra contains the Hecke operator $U_p$. We can then as usual define the ordinary projector as the idempotent $$e^{ord}=\lim\limits_{\xleftarrow{r}}D^{\vee}_{1,r}e^{ord}_t=\lim\limits_{\xleftarrow{r}}D^{\vee}_{1,r}\left(\lim\limits_{n\rightarrow\infty}U^{n!}_p\right).$$ Here $D^{\vee}_{1,r}$ denotes the dual of the degeneracy map defined above (which is a restriction to a specific set of cuspforms which are old at level $Np^{r+1}$) and $e^{ord}_r$ denotes the ordinary projector of level $\Gamma_0(Np^r)$.

While forming this space one constructs the universal Hecke algebra by taking the limit along system given by the restriction maps given by duality as the dual of the inclusions $$\bigoplus\limits_{k=1}^j S_k(\Gamma_0(Np^{r},\mathcal{O}_K)\rightarrow \bigoplus\limits_{k=1}^{j+1}S_k(\Gamma_0(Np^{r},\mathcal{O}_K),$$ $$S_k(\Gamma_0(Np^{r}),\mathcal{O}_K)\rightarrow S_k(\Gamma_0(Np^{r+1},\mathcal{O}_K).$$ It's important that for the second map the inclusion is $f(z)\mapsto f(z)$, not any other choice of degeneracy map so that $U_p$ maps to $U_p$. Since the hecke algebras acts continuously with respect to the norm, these construction agree and form a unique algebra of endomorphism on $S(\Gamma_0(Np^{\infty},\mathcal{O}_K)^{\mathfrak{p}\text{-adic}}$. This is the universal hecke algebra $\mathbb{T}(\Gamma_0(Np^\infty),\mathcal{O}_K)$.

As hinted at before the universal Hecke algebra contains the Hecke operator $U_p$. We can then as usual define the ordinary projector as the idempotent $$e^{ord}=\lim\limits_{n\rightarrow\infty}U^{n!}_p.$$

While forming this space one constructs the universal Hecke algebra by taking the limit along system given by the restriction maps given by duality as the dual of the inclusions $$\bigoplus\limits_{k=1}^j S_k(\Gamma_0(Np^{r},\mathcal{O}_K)\rightarrow \bigoplus\limits_{k=1}^{j+1}S_k(\Gamma_0(Np^{r},\mathcal{O}_K),$$ $$S_k(\Gamma_0(Np^{r}),\mathcal{O}_K)\rightarrow S_k(\Gamma_0(Np^{r+1},\mathcal{O}_K).$$ It's important that for the second map the inclusion is $D_{r,1}:f(z)\mapsto f(z)$, not any other choice of degeneracy map so that $U_p$ maps to $U_p$. Since the hecke algebras acts continuously with respect to the norm, these construction agree and form a unique algebra of endomorphism on $S(\Gamma_0(Np^{\infty},\mathcal{O}_K)^{\mathfrak{p}\text{-adic}}$. This is the universal hecke algebra $\mathbb{T}(\Gamma_0(Np^\infty),\mathcal{O}_K)$.

As hinted at before the universal Hecke algebra contains the Hecke operator $U_p$. We can then as usual define the ordinary projector as the idempotent $$e^{ord}=\lim\limits_{\xleftarrow{r}}D^{\vee}_{1,r}e^{ord}_t=\lim\limits_{\xleftarrow{r}}D^{\vee}_{1,r}\left(\lim\limits_{n\rightarrow\infty}U^{n!}_p\right).$$ Here $$D^{\vee}_{1,r}$$ denotes the dual of the degeneracy map defined above (which is a restriction to a specific set of cuspforms) and $e^{ord}_r$ denotes the ordinary projector of level $\Gamma_0(Np^r)$.

Fix a level $N$ and a prime $p$ such that $p\nmid N$. Just like in the classical case a normalized eigenform $f$ with integral Fourier coefficients (let's say they lie in the ring of integers $\mathcal{O}_F$ of a number ring $F$ or in its $\mathfrak{p}$-adic completion $\mathcal{O}_K$ at a prime $\mathfrak{p}$ above $p$) of weight $k$ and level $\Gamma_0(Np^r)$ defines a morphism of $\mathcal{O}_K$-algebras

Now this works perfectly fine on the ordinary part of universal Hecke algebra $\mathbb{T}(\Gamma_1(N),\mathcal{O}_K)^{ord}$, and it is actually somewhat the definition of Hida family given in Hida's 1986 paper. If this explains enough that's good! Otherwise below there is a more detailed explanation.

Fix a level $N$ and a prime $p$ such that $p\nmid N$. Just like in the classical case a normalized eigenform $f$ with integral Fourier coefficients (let's say they lie in the ring of integers $\mathcal{O}_F$ of a number ring $F$ or in its $\mathfrak{p}$-adic completion $\mathcal{O}_K$ at a prime $\mathfrak{p}$ above $p$, for now I will denote both of then by $\mathcal{O}$) of weight $k$ and level $\Gamma_0(Np^r)$ defines a morphism of $\mathcal{O}_K$-algebras

Now this works perfectly fine (up to an extension of scalars) on the ordinary part of universal Hecke algebra $\mathbb{T}(\Gamma_1(N),\mathcal{O}_K)^{ord}$, and it is actually somewhat the definition of Hida family given in Hida's 1986 paper. If this explains enough that's good! Otherwise below there is a more detailed explanation.