tRPC middleware infer type from another protectedProcedure

const studyMiddleware = middleware(async (opts) => {
  if (!opts.ctx.user) {
    throw new TRPCError({
      code: "UNAUTHORIZED",
      message: "Unauthorized",
    });
  }

  const parsedInput = z
    .object({
      studyId: z.string().cuid(),
    })
    .parse(opts.input);

  const study = await studyRepository.findOneByIdAndClientId({
    id: parsedInput.studyId,
    clientId: opts.ctx.user.client.id,
  });

  if (!study) {
    throw new TRPCError({
      code: "NOT_FOUND",
      message: "Study not found",
    });
  }

  return opts.next({
    ctx: {
      ...opts.ctx,
      user: opts.ctx.user,
      study,
    },
  });
});

export const budgetRouter = createTRPCRouter({
  create: protectedProcedure
    .use(studyMiddleware)
    .input(createBudgetInputSchema)
    .mutation(({ input, ctx: { user, study } }) =>
      budgetService.create(input, user, study),
    ),
});