Q:

ASP.Net Core Open Partial Generic Dependency Injection

I would like to register the following items for DI using an open generic implementation and interface. I know the following example will not work, as well as other combinations I've tried with MakeGenericType, or GetGenericArguments. I would like to simply call AddRepository<MyDbContext> and then be able to inject my implementation into classes without explicitly having to register the type I am using.
Interface
public interface IRepository<TEntity>
{   
}

Implementation
public class Repository<TEntity, TContext> : IRepository<TEntity>
    where TEntity : class
    where TContext : DbContext
{
}

Registration
public static class RepositoryServiceCollectionExtensions
{
    public static IServiceCollection AddRepository<TContext>(
        this IServiceCollection services) where TContext : DbContext
    {
        services.TryAddScoped(
            typeof(IRepository<>),
            typeof(Repository< , TContext>));

        return services;
    }
}

A:

The dependency injection container Microsoft.Extensions.DependencyInjection and its abstraction layer does not support open generic factories. So you generally cannot achieve what you would like to do there. There’s also no support planned.
Unlike many those other dependency injection related features, this is also not really possible to patch by just providing the right wrapper or factory types. So you will actually have to change your design here.
Since you want to resolve IRepository<TEntity> and the only way to do this is by registering an equivalent open generic type, you will have to have some type Repository<TEntity> that implements your repository. That makes it impossible to retrieve the database context type from the generic type argument, so you will have to use a different way here.
You have different options to do that. For example, you could configure your Repository<TEntity> (e.g. using M.E.Options) with the context type and make that resolve the Repository<TEntity, TContext> dynamically. But since you have actual control over your database context, I would suggest either adding a marker interface or introducing another type for the context which you can then register with the container:
public class Repository<TEntity> : IRepository<TEntity>
{
    public Repository(IDbContext dbContextFactory)
    { … }
}

public class MyDbContext : DbContext, IDbContext
{ … }

Then, your extension method could look like this:
public static IServiceCollection AddRepository<TContext>(this IServiceCollection services)
    where TContext : DbContext, IDbContext
{
    services.AddTransient(typeof(IDbContext), sp => sp.GetService<TContext>());
    services.TryAddScoped(typeof(IRepository<>), typeof(Repository<>));
    return services;
}

Of course, this changes how your Repository implementation works, but I don’t actually assume that you need to know the TContext type other than to inject the database context type. So this will probably still work for you.

That being said, I have too agree with Chris Pratt, that you probably don’t need this. You say that you want to introduce the repository, because “coding stores and implementations for every entity is a time consuming task” but you should really think about whether you actually need that. A generic repository is very limited in what it can do, and mostly means that you are doing just CRUD operations. But exactly that is what DbContext and DbSet<T> already do:

C: DbContext.Add, DbSet<T>.Add
R: DbContext.Find, DbSet<T>.Find
U: DbContext.Update, DbSet<T>.Update
D: DbContext.Remove, DbSet<T>.Remove

In addition, DbContext is a “unit of work” and DbSet<T> is an IQueryable<T> which gives you a lot more control and power than a generic repository could possible give you.

