How to maintain/prove authentication on cookie-enabled requests between HttpClient and asp core?

Question

Console app uses HttpClient to send basic auth request to asp core app.

HttpClient client = ...

client.DefaultRequestHeaders.Authorization = new AuthenticationHeaderValue(
"Basic", 
Convert.ToBase64String(
   encoding.GetBytes($"{user.UserName}:{user.Password}")));

var response = await client.SendAsync(request);

ASP Core app is configured as such (inline routing for simplicity):

public void ConfigureServices(IServiceCollection services)
{
    services.AddAuthentication(CookieAuthenticationDefaults.AuthenticationScheme).AddCookie();
}

public void Configure(IApplicationBuilder app, IWebHostEnvironment env)
{
    app.UseAuthentication();

    app.UseRouting(options =>
    {
        options.MapPost("/login", async (httpContext) =>
        {
            // get user creds
            var authHeader = httpContext.Request.Headers["Authorization"].First().Substring("Basic ".Length).Trim();

            ...

            // create claim and identity
            var claims = new List<Claim>
            {
                new Claim(ClaimTypes.Name, user.UserName)
            };
            var identity = new ClaimsIdentity(claims, CookieAuthenticationDefaults.AuthenticationScheme);

            // sign user in
            await httpContext.SignInAsync(CookieAuthenticationDefaults.AuthenticationScheme, new ClaimsPrincipal(identity));
        });
    });
}

When the client receives the response, there's a header called Set-Cookie with a value of something like ".AspNetCore.Cookies=foooooobarrrrrr; path=/; secure; samesite=lax; httponly"

• What am I supposed to do now on the client to ensure my future requests (from this client) are auth'd as this particular user?
• What do I need to include on followup requests to pass authentication?
• Does the server have to authenticate every request from here on and tie it back to this particular user??